ENDOCHRONIC THEORY OF PLASTICITY AT FINITE
DEFORMATIONS
Yulij Kadashevich, Sergei Pomytkin
Technological University of Plant Polymers Saint-Petersburg, Russia
ABSTRACT. Report is devoted to analysis of constitutive equations of endochronic
type at finite deformations. Some new parametric endochronic variants of theory are
presented. Methods of extension of endochronic theory of plasticity for finite
deformations region are proposed. The various incremental theories of plasticity are
initial. The variant-companion of endochronic theory is presented for each type of
incremental one. Against the classical endochronic theory, the equations in parametric
tensor form ( differential type ) are considered. The notions of reduced stresses,
reduced strains and its rates are introduced into constitutive relations for
generalization of endochronic theory on the finite deformations domain. Additionally
the original strain and stress measures adequate to the approach are proposed. The
new strain measure is compared with the wellknown ones on the simple shear test.
The effective method of tensor function calculations is presented. The method is
derived from formulae by Novozhilov and the common properties of matrix functions.
The possibilities of presented theory for applications are demonstrated for simple and
complex loading paths including the cyclic ones. Numerical simulations and
theoretical predictions are compared with experimental data.
ENDOCHRONIC THEORY FOR SMALL STRAINS
Against endochronic theory presented by Valanis (1, 2) the constitutive
equations of endochronic theory of plasticity in differential form can be written as (3)
 ij
2G

 d ij
2G dr

drij
drij

dr

rij
g 
(1.1)
or
 ij   
d ij

1 g
rij ,
g 
dr
dr
 ij
rij   ij  (1   )
dr  drij drij .
,
2G
Here  ij ,  ij , rij - deviators of stress tensor, strain tensor and parametric one,  endochronic parameter ( 0    1 ), G - shear modulus,  - analog of strain yield limit
(   0 ), g - analog of hardening coefficient ( g  0 ).
Obviously that if   1 then rij   ij , dr  d and relations have the view
 ij
d ij
 ij
 d ij
.


2G 2G d
d
g 1
It would be pointed that if   0 then rij   ijp , dr  d (  ijp - deviator of plastic
strain tensor,  - Odquist’s parameter ) and above mentioned endochronic theory is

incremental theory with linear Prager’s kinematical hardening.
At the same time another new endochronic equations of differential type can be used
in plasticity
3 – 105
 ij   
d ij
dr

drij

drij
dr

(1  g )
rij
(1  g )   g
or
 ij
2G
 ij

  d ij
2G dr
dr

rij
(1  g )   g
.
g
.
2G
1 g
For all variants of equations the curve “stress-strain” under uniaxial active loading
rij 
 (1   ) ij ,

dr  drij drij ,
tend to an asymtote
  (1  g )

2G
 g , g  0;   0 .
INTERCOUPLING INCREMENTAL AND ENDOCHRONIC TEORIES
Analysing various linear equations between stress and plastic strain tensors within the
framework of incremental theory with yield surface it was proved (3) that the highest
order of derivate of plastic strain tensor must be greater by unit than the highest
derivate of stress tensor. For example (  ij is deviator of backstress tensor,
     is tensor of active stress, ~    , a , b , k are constants),
ij
ij
ij
ij
 ij  ~
d  ijp
d
ij
i
i
 k ijp ;
or
 ij  ~
d  ijp
d
 ij  a1
  ij ,
d ij
d
 b1 ijp  b2
d ijp
d
;
or
 ij  a 2
d ij
b3  ijp
d  ijp
d 2  ijp

 b4
 b5
, etc.
d
d
d2
If the orders of derivates in left and right parts of are equal then the notion of yield
surface is vanished. Therefore adding the items with augmented order of stress
derivate to the left part of equation we can find the relation of endochronic type
(without yield surface and without loading-unloading condition). So any variant of
classical incremental theory of plasticity with isotropic and kinematical hardening has
its own variant-companion of endochronic theory. For example,
- incremental theory
d  ijp
~
 ij  
 k ijp
d
- endochronic theory-companion
~ d ij ~ drij
 ij 

 k rij .
2G dr
dr
More detail explanations can be found in (4).
3 – 106
ENDOCHRONIC THEORY AT FINITE DEFORMATIONS
In the process of extension of endochronic constitutive equations for finite
(large) deformations domain next statements will be foundation of approach:
a) at small strains the theory is obliged to transform to ordinary theory of endochronic
type;
b) oscillations of stress and deformations have to be absent in basic monotonic
loading;
c) stress and deformation tensors are indifferent;
d) constitutive relations of the theory are to satisfy to objectiivty principle.
Suppose that the motion of material point with material coordinates X 
(   1, 2, 3 ) into a position with the spatial coordinates x  (   1, 2, 3 ) is described
by the vector function x  x( X , t ) . (Further we omitt the lower indexes for tensors
but the upper sign “T” is operation of matrix transposition). The deformation gradient
x
is defined as F 
and can be polar decomposed into F  R u  v R . Here the
X
symmetric positive tensors u and v are the right and left stretch tensors, and the
proper orthogonal tensor R is the rotation tensor. Further, for the velocity gradient

L  F F 1 the following additive decomposition formulas can be used
1
1
L  D  W ; D  ( L  LT ) ; W  ( L  LT ) .
2
2
D
L
Stretching tensor
is the symmetric part of
and vorticiy tensor W is its skewsymmetric part. ( In the general case it would not confuse W with the time rate of R
and D with the derivative of strain measure tensor with respect to time).
Additionally, the obvious relations can be indicated u 2  F T F , R  F u 1 .
Using these notations, reduced stresses, reduced strains and its rates are
defined





T
E  R  R ; E  R DR ; T  R  R ; T  R  R ; N  R r R ; N  R r R .
Then these definitions are introduced into relations (1.1). We yield the endochronic
constitutive equations for finite deformations (     ):
T
T
T

r

T
T


 
T
1

T  N
r N,
2G 2G
g 





T
r  D  (1   )
, N  E  (1   )
2G
2G
Proposed equations satisfy the principles a)-d). In particular, it is well known that any
tensor A transformed according to formula A  R T A R is called indifferent or
independent of the observer.
MORE USUAL VIEW OF CONSTITUTIVE EQUATIONS
In order to apply the endochronic constitutive equations at finite deformations
in more comfortable and usual manner we realize inverse convolution transformation
by using operator R   R T . As result we yield
3 – 107






r

r  r



D
,





r  D  (1   )
2G



where for any tensor A we suppose that (  is named spin tensor)
r

2G
 
2G

g 

(4.1)

A  A A   A ,
  R RT .
It can be seen from these equations that if   1 then the relations have the simplest
form


D


D


   D 
2G
2G 1  g
.



D
If deformation rates are set then the equations are solved very simply. If stress rates
are assigned then the system are integrated a little more complex.
ON STRAIN MEASURE
According Hill’s proposals (5) a general class of Eulerian strain measures is
used in mechanics usually
3
3
u  k I
  f (u )   f (i )  
,
i 1
k 1 i   k
k i
where i are the distinct eigenvalues of symmetric positive definitive right stretch
tensor u . In particular, definition
un  I
n  0, 1, 2

,
n
is applied in practice. In this case n  0 defines the Hencky’s strain measure, n  1
assigns linear deformation and n  2 determines Green’s measure.
In fact the reduced strains and its rates

E  RT  R
E  RT D R
define a strain measure
t

  R   ( R T D R) dt  R T .
 0

In equivalent form relations (3) are a system of ordinary differential equations


      D ,
  R RT .
For simple shear when deformation gradient F and stretch rate D are
3 – 108
(5.1)
 1 2t 0 


F   0 1 0
 0 0 1


 0 1 0


D   1 0 0
 0 0 0


and
(5.2)
components of strain tensor, according (3), are defined by relations as
 11   22  cos 2   (2 ln cos  )  sin 2   (2   tg  )
.
 12  cos 2   (2   tg  )  sin 2   (2 ln cos  )
t  tg 
Strains calculated on basis of Green’s measure and Henky’ one are evaluated as
 11  0 ,  22  2t 2 ,  12  t
and
 11 
t ln 2
1 t2
 12 
,
ln 2
1 t 2
1  1,  2  1  t 2  t ,  3  1  t 2  t ,
respectively. Some numerical values of strains are presented in table 1. Here it was
accounted for that
 cos 

R    sin 
 0

sin 
cos 
0
0

0
1 
and
1

1 t2
 0 1 0


  1 0 0 .
 0 0 0


Table 1
Parameter
Measure (3)
Hencky's measure
t
11
12
11
12
0,5
0,208
0,435
0,215
0,431
1,0
0,571
0,693
0,623
0,623
5,0
2,140
3,333
2,267
0,454
10,0
3,126
7,830
2,983
0,300
SOME EXAMPLES
By tradition now consider the behaviour of stresses in simple shear of unit
cube. Using the constitutive equations (4.1) with   1 , 2G  1, 1 /( g   )  0,1 under
deformation gradient (5.2) we yield the stress response presented in Fig.1. (Simulation
was made in undimentional form).
3 – 109
Fig. 1
If the stress rate tensor



 0  0



 ij   0 0 
 0 0 0




is given then coordinates of motion vector can be looked like relations
 x1  a ( X 1  e X 2 )

.
 x2  bX 2
x  cX
3
 3
For this case deformation gradient and stretch rate are defined by tensors

 

ae
 a

0
 a
2b
 


 a a e 0


a e b

D
0 ,
and
F   0 b 0
b
 2b
0 0 c




c
0
 0

c



rotation tensor and spin one are determinated as
 cos sin  0 
 0 1 0
 



ae
tg  
and
.
R   sin  cos 0 
     1 0 0 ,
ab
 0



0
1

 0 0 0
The results of calculations of variables tg  , b and a are displayed in Fig.2. The
material constants were taken to be E  10 5 MPa ,   0.25 ,  0  207 MPa ,   1 ,
3 – 110
1  g  0.5 . The data of this numerical experiment has the good agreement with
relults published in (6).
Fig. 2
Additionally it would be noted that under the cyclic deformation in simple
shear conditions the Baushinger effect is described. Its response and magnitude don’t
depend on strain rate and endochronic parameter.
Analysis of some another spin tensors and stress behaviour in the frameworks
of incremental theory can be found, for example, in (7).
ON MATRIX FUNCTIONS
In the processes analytical and numerical simulation of finite deformations
there is a need to calculate matrix functions very often. The basic method of
calcucation for matrix a its matrix function f (a ) is well-known
3
3
(a  k I )
f ( a )   f ( i )  
.
i 1
k 1 (i   k )
k i
Here I is unit tensor, 1 , 2 ,  3 are eigen-values of tensor a which are found from
cubic equation det (a   I )  0 usually. In two-dimensional case (when
a13  a23  a31  a32  0 ) eigen-value 1 is equal to a33 always and residuary
quadric equation is solved easy. But in the general case the solving of cubic equation
can be laborious. Using the Novozhilov’s formulae (8)
J
2
2  J
2

2 
J 2  sin       1 ,
1 
J 2  sin   1 ,
3
3  3
3
3

3 
J 1  aii ,
J 2  aij a ji ,
2
4  J

J 2  sin       1 ,
3
3  3

J 3  aij a jk a ki ,
sin 3  
6 J3
3
J 22
the problem can be simplified.
ACKNOWLEDGMENTS
The research described in this publication was made possible in part by Grant
No. 03-01-00770 from Russian Foundation for Basic Researches and Grant No. E-024.0-158 from Ministry of Education of Russian Federation.
3 – 111
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